Dynamical models to explain observations with SPHERE in planetary systems with double debris belts

International audienceContext. A large number of systems harboring a debris disk show evidence for a double belt architecture. One hypothesis for explaining the gap between the debris belts in these disks is the presence of one or more planets dynamically carving it. For this reason these disks represent prime targets for searching planets using direct imaging instruments, like the Spectro-Polarimetric High-constrast Exoplanet Research (SPHERE) at the Very Large Telescope. Aim. The goal of this work is to investigate this scenario in systems harboring debris disks divided into two components, placed, respectively, in the inner and outer parts of the system. All the targets in the sample were observed with the SPHERE instrument, which performs high-contrast direct imaging, during the SHINE guaranteed time observations. Positions of the inner and outer belts were estimated by spectral energy distribution fitting of the infrared excesses or, when available, from resolved images of the disk. Very few planets have been observed so far in debris disks gaps and we intended to test if such non-detections depend on the observational limits of the present instruments. This aim is achieved by deriving theoretical predictions of masses, eccentricities, and semi-major axes of planets able to open the observed gaps and comparing such parameters with detection limits obtained with SPHERE. Methods: The relation between the gap and the planet is due to the chaotic zone neighboring the orbit of the planet. The radial extent of this zone depends on the mass ratio between the planet and the star, on the semi-major axis, and on the eccentricity of the planet, and it can be estimated analytically. We first tested the different analytical predictions using a numerical tool for the detection of chaotic behavior and then selected the best formula for estimating a planet's physical and dynamical properties required to open the observed gap. We then apply the formalism to the case of one single planet on a circular or eccentric orbit. We then consider multi-planetary systems: two and three equal-mass planets on circular orbits and two equal-mass planets on eccentric orbits in a packed configuration. As a final step, we compare each couple of values (Mp, ap), derived from the dynamical analysis of single and multiple planetary models, with the detection limits obtained with SPHERE. Results: For one single planet on a circular orbit we obtain conclusive results that allow us to exclude such a hypothesis since in most cases this configuration requires massive planets which should have been detected by our observations. Unsatisfactory is also the case of one single planet on an eccentric orbit for which we obtained high masses and/or eccentricities which are still at odds with observations. Introducing multi planetary architectures is encouraging because for the case of three packed equal-mass planets on circular orbits we obtain quite low masses for the perturbing planets which would remain undetected by our SPHERE observations. The case of two equal-mass planets on eccentric orbits is also of interest since it suggests the possible presence of planets with masses lower than the detection limits and with moderate eccentricity. Our results show that the apparent lack of planets in gaps between double belts could be explained by the presence of a system of two or more planets possibly of low mass and on eccentric orbits whose sizes are below the present detection limits. Based on observations collected at Paranal Observatory, ESO (Chile) Program ID: 095.C-0298, 096.C-0241, 097.C-0865, and 198.C-0209